Goals: - To extend the theory of topological coverings to that of holomorphic (complex analytic) coverings - To show that any Riemann surface structure on the base space of a topological covering induces a Riemann surface structure on the covering space in such a way that the covering projection map is holomorpic. To achieve this using the technique of "pulling back charts from below" - To see why the Riemann surface structure induced above is essentially unique - In particular, we get a unique Riemann surface structure on the topological covering of a Riemann surface. The deck transformations therefore become holomorphic automorphisms of this Riemann surface structure Topics: Topological covering, holomorphic covering, admissible neighborhood, chart, pulling back charts by local homeomorphisms, locally biholomorphic, pulling back Riemann surface structures, holomorphicity or complex analyticity of continuous liftings, deck transformation
The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmueller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra---especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory. The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given.